``Generalized Sequential Monte Carlo Sampling for Redistricting Simulation.''

Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.

McCartan, Cory and Kosuke Imai. (2023). ``Sequential Monte Carlo for Sampling Balanced and Compact Redistricting Plans.'' Annals of Applied Statistics, Vol. 17, No. 4 (December), pp. 3300-3323.

Kenny, Christopher T., Cory McCartan, Tyler Simko, Shiro Kuriwaki, and Kosuke Imai. (2023). ``Widespread Partisan Gerrymandering Mostly Cancels Nationally, but Reduces Electoral Competition .'' Proceedings of the National Academy of Sciences, Vol. 120, No. 25, e2217322120.

McCartan, Cory, Christopher T. Kenny, Tyler Simko, George Garcia III, Kevin Wang, Melissa Wu, Shiro Kuriwaki, and Kosuke Imai. (2022). ``Simulated redistricting plans for the analysis and evaluation of redistricting in the United States.'' Scientific Data, Vol. 9, No. 689, pp. 1-10.

Fifield, Benjamin, Michael Higgins, Kosuke Imai, and Alexander Tarr. (2020). ``Automated Redistricting Simulator Using Markov Chain Monte Carlo.''Journal of Computational and Graphical Statistics, Vol. 29, No. 4, pp. 715-728.

Fifield, Benjamin, Kosuke Imai, Jun Kawahara, and Christopher T. Kenny. (2020). ``The Essential Role of Empirical Validation in Legislative Redistricting Simulation.'' Statistics and Public Policy, Vol. 7, No. 1, pp 52-68.

Fifield, Benjamin, Christopher T. Kenny, Cory MaCartan, Alexander Tarr, and Kosuke Imai. ``redist: Computational Algorithms for Redistricting Simulation.'' available through The Comprehensive R Archive Network and GitHub.